US8189900B2 - Image-based methods for measuring global nuclear patterns as epigenetic markers of cell differentiation - Google Patents
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Abstract
Description
may be used to smooth and segment the image. Equations useful to solve for smoothed data u and an edge function v may comprise:
This energy functional is minimized with respect to u and v, where u(x) is a vector scalar field, and v(x) is a vector or scalar edge strength fields, and both are defined in Rn. The scalar or vector g is the observed input data, ƒ(u) is generally a function of u with respect to which the segmentation is desired, and h(u, g) is the measurement model. The notation ∥z∥2 for an arbitrary vector z represents the Euclidean norm. The terms ƒx(u) and vx represent, respectively, the gradient of ƒ and v with respect to the spatial variable x. The constants α, β and ρv represent scalar weights, which balance the various effects. v takes a value between 0 and 1. As such, it can be interpreted as the probability that the particular pixel is an edge pixel. Note that if v is 1, the first term in the integrand vanishes, meaning that no smoothing needs to occur across this pixel since it is an edge pixel; this is the key to segmentation. The third term smoothes the edge function itself, while the fourth term penalizes edges so as to avoid data overfitting.
h(u,g)=∥u−g∥ 1 (2)
where ∥z∥1 represents the 1-Norm of z (Desai, Kennedy, Mangoubi, et al. 2002). The 1-Norm is particularly appropriate when the measurement is corrupted by fatter tail noise as found in images, such as Laplacian noise. This norm is also appropriate when the data contains abrupt changes, such as edges. Finally, when ƒ is the identity function then we are simply smoothing the gradient of u. If u is a vector function, then ƒ(u) can be a norm of this vector.
where M, p, are parametric functions, Lm, Cu, and Lc are spatial operators. The energy functional E to be minimized given data g has now 7 scalar or vector arguments. The first two, u, and v, are well known from fMRI (Desai, Mangoubi, 2002 & 2006, enclosed) and DWI (Desai, Kennedy, Mangoubi, et al, 2006); they represent the smoothed image and the associated edge fields, respectively. The additional variables represent the measurement and image model characteristics. Specifically, θm and vm, represent the measurement model parameters and its edge field respectively, while, θu, and vu represent the image model parameters and its edge field, respectively. The data fidelity and correlation model parameter continuity expressions (1st and 3rd term) can now depend on unknown parameters to be learned, θm and θu. For these, we impose the two learning models, Lm and Lc (2nd and 4th term). Finally, the last term incorporates prior information. Priors on edge can also include information on shape future and orientation. The energy functional above may enable learning in the context of (1) the measurement function embodied in the data fidelity term M(u,g,w,θm) for both scalar and vector measurements g, (2) the spatial correlation operator of the process u, embodied in Cu(u,θu,vu) applicable to scalar and vector input data, and (3), the spatial correlation operators Lm, Lc, for the parameters, θm and, θu, respectively. In previous segmentations effort, the stationarity of the texture field over the area of interest has been assumed. The new formulation may handle adaptive learning of the spatially varying textures as well as segmentation of regions of homogeneity of texture, where the variations may be due to noise or the texture itself. As such, it permits the adaptation of both the neighborhood size and shape throughout texture at each pixel throughout the image.
s 2=(x 2 ,y 2)=(x 1 +d x ,y 1 +d y)=S 1 +d (2.5)
For a fixed displacement d, statistical methods assume that the probability that s1 and s2 take on grayscale values of i and j, respectively, is governed by the joint probability mass function (PMF) P(i, j; d). We may equivalently reference the separation displacement between s1 and s2 by an absolute distance d and angle θ relative to the horizontal axis. The PMF of the spatial grayscale values becomes in this case P(i, j; d, θ).
TABLE 2.1 |
Haralick's Statistical Texture Features |
Textural Feature Name | Formula | |
Angular Second Moment | ΣiΣj ({circumflex over (P)}(i, j: d, θ))2 | |
Contrast |
|
|
Correlation |
|
|
Entropy | −ΣiΣj {circumflex over (P)}(i, j; d, θ) log ({circumflex over (P)}(i, j: d, θ)) | |
for p=1, 2, . . . Nx and q=1, 2, . . . , Ny where j is the imaginary number square root of −1. The Fourier coefficients F1(p, q), or some energy measures extracted from them can then be used as textural features.
F w(u,ξ)=∫−∞ ∞ƒ( x)w(x−ξ)e −j2πux dx (2.7)
where u and ξ represent the frequency and spatial components of the transformed signal Fw(u,ξ). Equation 2.7 becomes the Gabor transform when the window function w(.) is Gaussian. Applying similar ideas to a two-dimensional image yields the two-dimensional Gabor filter method. For example, the impulse response of an even-symmetric (i.e., real part only) two-dimensional Gabor filter with a 0° orientation to the horizontal axis is,
where u0 is the frequency along the horizontal axis and σx, and σy govern the Gaussian envelope (window) that modulates the frequency sinusoid. Different orientations can be analyzed via axis rotation of the x and y coordinates.
wavelet representation. (G. Van de Wouwer, P. Scheunders, and D. Van Dyck. Statistical texture characterization from discrete wavelet representation. IEEE Transactions on Image Processing, 8(4):592-598, April 1999).
is the mean value of the coefficients at the both subband. Similarly, the mean deviation, or absolute mean, is defined as,
Applying Equations 2.10 and 2.12 across all B subbands in the wavelet decomposition of an image yields the wavelet energy signature of that image. Thus, the vector,
x=[E 1 , E 2 , . . . , E B , MD 1 , MD 2 , . . . , MD B] (2.13)
is the resulting wavelet energy textural feature representation.
x i =Sθ i +n i i=1 . . . I (4)
where xi is the Ith measurement, S is the subspace or data representation of interest, θi is the intensity of the signal of interest in the Ith measurement, and ni is the noise contribution. If the noise is Gaussian, then conventional subspace estimation methods, such as principal component analysis (PCA), may be used. In some embodiments, we expect the noise to be non-Gaussian. For non-Gaussian noise, we have formulated a Generalized Likelihood Ratio (GLR) approach as follows
where p(xi−Sθi)=p(ni) is the probability density function of the noise. In a recent study (Desai and Mangoubi 2004 and 2004-2), the necessary optimality conditions have been derived for the above optimization problem for a large class of noise density functions.
where Δ is a measure of the separation achieved between the two classes. Maximizing this separation measure Δ can yield a smaller VC-dimension, which in turn yields a tighter upper bound on the generalization error. The support vector machine creates just such a maximally separating hyperplane.
SVM Formulation
w·x i +b≧1 ∀i such that y i=+1 (2.16)
w·x i +b≦−1 ∀i such that y i=−1 (2.17)
where w ∈ d, b ∈ is a scalar bias term (Osuna, 1998) We can write the above expressions more compactly as,
y i(w·x i +b)≧1 ∀i=1, 2, . . . , N (2.18)
(Edgar Osuna. Support Vector Machines: Training and Applications. PhD thesis, Massachusetts Institute of Technology, 1998). This expression represents the constraint that all training points must be correctly classified by the hyperplane.
If we impose the normalization mini=1 . . . N|w xi+b|=1, the distance from the hyperplane to the nearest point of either class, or the margin, is simply
In order to maximize this margin while ensuring that all points are correctly classified according to Equation 2.18, the following optimization problem may be posed:
(Edgar Osuna. Support Vector Machines: Training and Applications. PhD thesis, Massachusetts Institute of Technology, 1998), Noting that maximizing ∥W∥ is equivalent to minimizing ½∥w∥2, we have the following equivalent quadratic optimization formulation:
where λi is the Lagrange multiplier associated with the training point xi, K(., .) is a kernel function and C is a cost penalty. Solving this quadratic optimization problem returns the λi values that defines the maximally separating hyperplane.
K(x i ,x j)=(x i ·x j+1)p (2.23)
and the radial basis function kernel with width σ
K(x i ,x j)=e −∥x
is the Gamma function defined on z>0. The two GGD parameters α and β control the width and decreasing rate of the distribution's peak. The GGD reduces to the Laplacian and Gaussian distributions for β=1 and β=2, respectively. Minh N. Do and Martin Vetterli. Wavelet-based texture retrieval using generalized Gaussian density and Kullback-Leibler distance. IEEE Transactions on Image Processing, 11(2):146-158, February 2002. In fact, as β→∞, the GGD approaches the uniform distribution. As described, GGD is a general, unimodal and symmetric distribution method able to model a large range of data through the estimation of two parameters.
is the Kullback-Leibler distance operator at the both subband (Minh N. Do and Martin Vetterli. Wavelet-based texture retrieval using generalized Gaussian density and Kullback-Leibler distance. IEEE Transactions on Image Processing, 11(2):146-158, February 2002).
That is, with the independence assumption, one can sum up the KLD's across all subbands.
are the Kullback-Leibler distances between the test point's wavelet coefficient distribution and the +1 and −1 classes' wavelet coefficient distributions, respectively. Then one can write Equation 2.33 as the KLD classification function,
ƒ(x)=sign(KLD(x,−1)−KLD(x,+1)) (2.36)
Addtionally, we introduce a KLD confidence measure,
h(x)=KLD(x,−1)−KLD(x,+1) (2.37)
Texture Classification: The Likelihood Ratio Test and its Probability Density Function
where x is the random variable for a particular wavelet subband of the texture, w is a width parameter proportional to the standard deviation, and p is a tail thickness shape parameter. A third parameter, the location or mean, is found to be zero in (1) for the subbands of interest. The term F(1/p) represents the Gamma function. Note that when (p, ω)=(2, √2), we have a standard Gaussian density. Each texture would have a representative density function (1) at each decomposition level. Thus, given a texture decomposition that provides B wavelet bands, the characterizing pdf for that texture is
That is, given a new texture that needs to be assigned to one class from among C classes, we compute the wavelet coefficients from samples for the new texture, and we use these sample decompositions to estimate the generalized Gaussian parameters. We now have knowledge of the new texture's pdf, ƒnew, and can select the class c* whose pdf has the shortest KL distance from the new texture's pdf.
where each of the densities p0b and p1b, b=1, . . . , B is a generalized Gaussian density function given in (1), with respective parameters (ω1, p1), . . . , (ωB, pB).
For p=2, we have the χ2 random variable, which is the square of a standard normal variable. Likewise, χp is a generalization of χ2 with respect to the generalized Gaussian variable with width parameter ω=1, raised to the power p, whose pdf is
The χ2 N, or the χ2 random variable with N degrees of freedom, is simply the sum of N independent χ2 random variables. Likewise, we can define the χp N random variable to be the sum of N independent χp random variables.
The log-likelihood ratio for the hypothesis test (5-6) is expressed in terms of these new random variables. With N=SB, where S is the number of samples for the texture and B the number of wavelet bands per sample, we have
where K is a constant dependent on the scale and shape parameters. Now we have the test
where χ(p,N) is a Generalized Gaussian-based generalization of the Chi Square (when p=2) statistic with N degrees of freedom, and Tc(p1, p2) is a threshold for significance level c. As with the Chi Square variable, χ(p, N) is the sum of N one degree of freedom χ(1, p) variables. These have unnormalized density functions
ƒ(χ(p,1))=exp(−χ(p,1))(χ(p,1))(1-p)/p (19)
When p=2, we have the un-normalized Chi Square density. The probability density for the likelihood ratio (1) can thus be obtained by convolving numerically densities of the form (2) (see Desai and Mangoubi, 2003, for the difference between two Chi Square variables with different degrees of freedom). By numerically integrating the resultant density of the log-likelihood ratio (1), we obtain the critical values for significance levels without Monte Carlo runs.
p ij(k+1)=a ij p ij(k)+σij r ij(k) (21)
where aij and σij are the band width and the standard deviation (or spectral height), respectively, and rij(k) is a white noise process with unit standard deviation. Similar models were estimated for the width parameters wij. This model is a scalar discrete time and stochastic version of Equation 24, described below.
dx/dt=A i x(t)+B jd i u(t)+G i w(t) (22)
where x(t) represents the features of interest at time t, u(t) is the media input, and w(t) is random noise. The subscript i represents the state of differentiation. For instance, if i=1, then the nucleus is pluripotent. If i=2, 3, . . . , it is differentiated to various degrees. Intermediate stages are also possible. This time varying dynamic system can help predict the evolution of different features such as connectivity and nuclei shape deformation within each phase. Note that the image observation data are a function of the state x(t). Thus, we have at time tk an image representation of the form
y(tk)=hi(x(tk))+v(tk) (23)
where v(tk) is an observation noise. In some embodiments, the observation in Eq. (11) may be used to estimate in real time the evolution of Eq. (10). Kalman filtering (Poor 1998, Mangoubi 1998) may be used, and if the model Eq. (10) is not adequately known, then more advanced robust filtering (see, e.g., Mangoubi 1998) may be appropriate. At the higher level, the transition between phases may be modeled as a discrete state continuous time Markov Chain. Thus, we have the Hidden spatio-temporal Markov Model,
dp(t)/dt=Tp(t) (24)
where p(t)=[p1 . . . , pI] is the probability vector for each state i at time t, and T is the transition probability matrix. In some embodiments, the transition probability matrix is a function of the environmental conditions. Also, since differentiation is a unidirectional process, T is upper triangular.
Bilinear model: dx(t)/dt=c 1(1−x(t))u(t)−c 2 x(t)
Linear model: dx(t)/dt=c 1 u(t)−c 2 x(t)
where c1 and c2 are positive constants and u is an input function such as environmental conditions. The first model is bilinear where the input function multiplies the state, and we have saturation at steady state. In the second model, we have a linear model that does not saturate at steady state. Specifically, at steady state, we have
Bilinear model: x ss =c 1 u/(c 2 −c 1 u)
Linear model x ss =c 1 u/c 2
In brief, bilinear models can predict the saturation effect due to the finite capacity of the medium and interacting entities over finite space. These models have successfully predicted the behavior of chemical sensors in finite capacity interactive setup environments.
-
- 1. Compute the divergence from the unclassified window to each of the library windows.
- 2. Select the k library windows with the smallest distance to the unclassified window (i.e., the nearest neighbors) and allow them to “vote” on its class.
- 3. Assign the unknown window to the class which receives the most “votes.”
TABLE I |
Typical Classification Accuracy; |
classification parameters: k = 7, kn = 5. Daubeckies-4 wavelet; |
window size is 64 × 64 |
Accuracy |
90% Conf. Int. | ||
Pluripotent | 0.996 | [0.9812, 0.9986] |
Differentiated | 0.892 | [0.8547, 0.9196] |
Exterior | 0.968 | [0.9432, 0.9812] |
TABLE II |
Typical Misclassitied Windows, |
Sample Size is 250: classification parameters: k = 7, kn = 5, |
Daubechies-4 wavelet, window size is 64 × 64 pixels |
Errors | Pluri. | Diff. | Ext. | | |
Pluripotent |
1 | — | 0 | 1 | 0 | |
Differentiated | 27 | 0 | — | 19 | 8 |
|
8 | 5 | 3 | — | 0 |
TABLE III |
Robustness to Variation in Classification Parameters: window size is 64 × 64 pixels. Danbechies-4 wavelet |
Pluripotent | Differentiated | Exterior |
k | kn | Accuracy | 90% Conf. Int. | |
90% Conf. Int. | |
90%, Conf. Int. |
1 | 1 | 0.996 | [0.9812. 0.9986] | 0.900 | [0.8637. 0.9264] | 0.952 | [0.9237. 0.9691] |
3 | 3 | 0.996 | [0.9812, 0.9986] | 0.808 | [0.8282. 0.8987] | 0.964 | [0.9383. 0.9782] |
3 | 5 | 0.992 | [0.9751. 0.9967] | 0.908 | [0.8727. 0.0332] | 0.960 | [0.9334. 0.9752] |
5 | 7 | 0.996 | [0.9812, 0.9986] | 0.892 | [0.8547, 0.9196] | 0.968 | [0.9432, 0.9812] |
TABLE IV |
Robustness to Variation in Modeling Parameter (Mother Wavelet): window size is 64 × 64 pixels. k =7, kn = 5 |
Pluripotent | Differentiated | Exterior |
Wavelet | | Accuracy | 90% Conf. Int. | | 90% Conf. Int. | | 90% Conf. Int. | |
Daubechies-2 | 4 | 0.976 | [0.9534, 0.9868] | 0.952 | [0.9237. 0.9691] | 0.976 | [0.9534. 0.9868] |
Symlet-2 | 4 | 0.976 | [0.9534, 0.9868] | 0.952 | [0.9237. 0.9691] | 0.976 | [0.9534. 0.9868] |
Biorthogonal 1.3 | 6 | 0.984 | [0.9639. 0.9921] | 0.892 | [0.8547, 0.9196] | 0.972 | [0.9483, 0.9840] |
Coiflet-1 | 6 | 1.000 | — | 0.912 | [0.8772, 0.9366] | 0.972 | [0.9483, 0.9840] |
Reverse Biorthogonal 1.3 | 6 | 1 .000 | — | 0.908 | [0.8727, 0.9332] | 0.972 | [0.9483. 0.9840] |
Daubechies-4 | 8 | 0.996 | [0.9812, 0.9986] | 0.892 | [0.8547, 0.9196] | 0.968 | [0.9432. 0.9812] |
Symlet-4 | 8 | 0.996 | [0.9812, 0.9986] | 0.916 | [0.8818, 0.9399] | 0.972 | [0.9483. 0.9840] |
Biorthogonal 2.4 | 10 | 0.992 | [0.9751. 0.9967] | 0.848 | [0.8063. 0.8810] | 0.964 | [0.9383. 0.9782] |
Reverse Biorthogonal 2.4 | 10 | 0.992 | [0.9751, 0.9967] | 0.868 | [0.8282, 0.8987] | 0.964 | [0.9383, 0.9782] |
Coiflet-2 | 12 | 0.984 | [0.9639, 0.9921] | 0.856 | [0.8150. 0.8881] | 0.956 | [0.9285. 0.9722] |
Daubechies-8 | 16 | 0.980 | [0.9586, 0.9895] | 0.848 | [0.8063. 0.8810] | 0.956 | [0.9285. 0.9722] |
Segmentation of Classified Images
H 0:θ<θ0
H 1:θ>θ1 (29)
ƒ(x i/θj)=θ1 x
TABLE V |
Expected SPRT Test Length; θ0 = 0.8. θ1 = 0.95. |
to the left of the slash is expected test length |
assuming H0, to the right assuming H1 |
β |
α | 0.05 | 0.10 | 0.15 | |
0.05 | 18.96/28.21 | 14.27/26.83 | 11.53/25.70 | |
0.10 | 17.00/20.03 | 12.58/18.71 | 10.01/17.64 | |
0.15 | 15.25/15.19 | 11.09/13.93 | 8.69/12.93 | |
Antigen | Location | Antibody | Source | Number | Dilution |
CREST | Centro- | Human | Gift-Dr. Cal | N/A | |
meres | polyclonal | Simerly | |||
H3 acetyl | Euchro- | Rabbit | Upstate | 06-599 | 1:200 |
lys9 | matin | polyclonal | Biotech | ||
H3 TriMe | Hetero- | Rabbit | Upstate | 07-030 | 1:100 |
lys9 | chromatin | polyclonal | Biotech | ||
lamin A/C | Nuclear | Rabbit | Santa Cruz | sc-20681 | 1:100 |
laminae | polyclonal | ||||
nestin | Neural | Rabbit | Ab Cam | AB 7659 | 1:100 |
progenitor | polyclonal | ||||
nuclear | Nuclear | Mouse | Covance/ | MAb414 | 1:250 |
porins | laminae | monoclonal | Babco | ||
Oct-4 | Pluriotent | Mouse | Santa Cruz | 1:100 | |
stem cells | monoclonal | ||||
Pax-6 | Neural | Rabbit | Covance | PRB 278 P | 1:200 |
progenitor | polyclonal | ||||
Time Lapse Confocal Microscopy:
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